Automated certified analysis of dynamic systems AFOSR FA

Automated, certified analysis of dynamic systems AFOSR: FA 9550 -05 -1 -0266, 2005 -08 (PM: Lt. Col Scott Wells, Ph. D. ) NASA: 2008 -11. (PM: Dr. Christine Belcastro) – (related) NSF: CTS-0113985 (2001 -05) and CHE-0535542 (2006 -10) Participants UCB: Ufuk Topcu, Weehong Tan, Packard, Frenklach, Feeley, Russi, Wheeler Honeywell: Pete Seiler UMN: Gary Balas More info http: //jagger. me. berkeley. edu/~pack/certify http: //jagger. me. berkeley. edu/~pack/certificates http: //jagger. me. berkeley. edu/~pack/nsfuncertainty Copyright 2007, Packard, Frenklach, Topcu, Feeley, Russi, Tan, Wheeler, Seiler and Balas. This work is licensed under the Creative Commons Attribution-Share. Alike License. To view a copy of this license, visit http: //creativecommons. org/licenses/by-sa/2. 0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. index

Our Perspective Summary Nonlinear Analysis Region-of-Attraction Disturbance-to-Output (L 2→L∞) gain Disturbance-to-Output (L 2→L 2) gain Set Containment condition Sample Problems Algebraic Tools SOS Decomposition Semi. Definite Programming (SDP) Sum-of-Squares (SOS) as SDP Synthesizing SOS polynomials Dimensions Bilinear SOS problems Psatz Region of Attraction Lyapunov, enlarge estimate Convexity of analysis BMI formulation, Initial Algorithms, simple example Simulations, Outer Bound for V, Sampling Certifying with a given V Summary Uncertainty, general discussion Affine uncertainty, Branch & Bound, 2 -state example index ROA: Examples Van der. Pols Aircraft ROA (with uncertainty) ROA w/ delay (with uncertainty) Covering non-affine uncertainty Reachability Linear, Nonlinear as bilinear SOS Lower Bound Refinement Input/Output Gain Adaptive control example SOS conditions Results Recap, Impact, Transitions, Discussion

Our Perspective Current Practice – Linear analysis: Provides a quick answer to a related, but different question: Q: How much gain and time-delay variation can we accommodate in flight? A: Here’s a scatter plot of gain margin/time-delay margin at 1000 trim conditions (throughout envelope) – Why does linear analysis have impact in nonlinear problems? • Domain-specific expertise exists to interpret linear analysis & assess relevance • Speed, scalable: Fast, defensible answers, on high-dimensional systems Questions: • To what extent is V&V adequately addressed with current approach of linear analysis plus nonlinear simulation? Or… • Should nonlinear analysis be a major part of V&V? • What are appropriate stability margin metrics to use for nonlinear analysis? index

Tools for Quantitative, Local Nonlinear Analysis Focus index Comparison to Literature – Region-of-attraction estimation – induced norms for locally stable, finite-dimensional nonlinear systems, with – polynomial vector fields – parameter uncertainty (also polynomial) Main Tools: Lyapunov, with – Sum-of-squares proofs to ensure nonnegativity and set containment – Semidefinite programming (SDP), Bilinear Matrix Inequalities (BMI) – Only method to incorporate both simulation and certificates of stability – Superior to other general purpose methods Doing examples, gaining experience – 2, 3, 4, 5 and 6 -d examples – http: //jagger. me. berkeley. edu/~pack/certificates – Simulation: practical, informative, does aid the search for Lyapunov functions to certify an ROA Pragmatic Goal – A feasible path towards attacking problems with, eg. , 15 states, 5 uncertainties, and cubic (in state) vector fields • Interface: @polysys, SOStools, YALMIP • SDP solvers: Sedumi • BMIs: PENBMI – Constraints provided by simulation – Parameter-Independent Lyapunov functions (“quadratic stability” from late 80’s) – Branch & Bound in uncertainty space Payoff: quantitative analysis where – Insufficient domain-specific knowledge • no experience to rely on linear analysis – performance is being pushed to the limit • approximations associated with linear analysis are not suitable

Nonlinear Analysis Autonomous dynamics – equilibrium point – uncertain initial condition, – Question: do all solutions converge to Driven dynamics – equilibrium point – uncertain inputs, , – Question: how large can Uncertain dynamics – Unknown, constant parameter, – Same questions… get? index

Uncertain ICs: Estimating Region of Attraction Dynamics, equilibrium point If there exists positive-definite V with then (Lyapunov): the set and in the region-of-attraction. is invariant, Strategy: Grow estimate via optimization over V, enforcing containments with S-procedure/SOS conditions index

Reachability of with inputs index If there exists positive-definite V with Conditions on then Similarly, if then Conclusion on ODE

gain of If there exists positive-definite V with then Similarly, if then index

S-procedure: Checking Set Containments Consider sets defined via inequalities, using given functions f. A and f. B as Question: Does the set containment constraint hold? If there exists a nonnegative function s(x) such that for all then But… –Checking that a function is globally nonnegative is hard (in general) –Even hard to verify that a polynomial in many real variables is nonnegative –Searching over globally nonnegative functions (polynomials) is hard How can this simple approach be used? index

Uncertain System Stability Analysis Pitch-Axis Aircraft Model: – Short period longitudinal model, with parameter uncertainty – 2 -state dynamic inversion controller Task: Prove (or disprove) that – for all values of δ 1 and δ 2 satisfying -0. 1≤ δi ≤ 0. 1, and – for all initial conditions, z(0)=z 0, η(0)= η 0 satisfying Perhaps change 10 to β and maximize β so statement is true the solution of the governing ODEs exists and decays to 0 (the equilibrium point) index

Uncertain Model Invalidation Analysis Given time-series data for a collection of experiments, with selected features and simple measurement uncertainty descriptions… Task: prove that regardless of the values chosen for the parameters, the model below cannot account for the observed data, where index

Sum-of-Squares index Sum-of-squares decompositions will be the main tool to decide set containment conditions, and certify nonnegativity. A polynomial f, in n real-variables is a sum-of-squares if it can be expressed as a sum-of-squares of other polys, Obviously, SOS polynomials are globally nonnegative Notation set of all sum-of-square polynomials in n variables set of all polynomials in n variables

Semidefinite programming Given symmetric matrices and , and Generalization of linear programming Convex feasable set Good complexity, duality theory Available algorithms, active OR research (sparsity, structure) Se. Du. Mi, GNU license Workhorse in control for past 15 years Bilinear version: not convex, local algorithms, arises naturally in places index

Sum-of-Squares Decomposition index For a polynomial f, in n real-variables, and of degree 2 d The entries of z are not algebraically independent – e. g. x 12 x 22 = (x 1 x 2)2 ) – M is not unique (for a specified f) The set of matrices, M, which yield f, is an affine subspace – one particular + all homogeneous – Particular solution depends on f – all homogeneous solutions depend only on n & d. Searching this affine subspace for a p. s. d element is an SDP…

Sum-of-Squares as SDP For a polynomial f, in n real-variables, and of degree 2 d Semidefinite program: feasibility Each Mi is s×s, where Using the Newton polytope method, both s and q can often be reduced, depending on the terms present in f. index

index (s, q) dependence on n and 2 d 2 d n 2 4 6 8 2 3 0 6 6 10 27 15 75 3 4 0 10 20 20 126 35 465 4 5 0 15 50 35 420 70 1990 6 7 0 28 196 84 2646 210 19152 8 9 0 45 540 165 10692 495 109890 10 11 0 66 1210 286 33033 1001 457743

Synthesizing Sum-of-Squares as SDP Given: polynomials Decide if an affine combination of them can be made a sum-of-squares. This is also an SDP. index

Checking Set Containments Using SOS index Consider sets defined via inequalities, using given polynomials p. A and p. B as Question: Does the set containment constraint If there exists a polynomial such that then Linearly parametrize class to search (for s) over as Find so hold?

Psatz Given: polynomials Goal: Decide if the set is empty. Φ is empty if and only if such that index

Synthesizing Sum-of-Squares as Bilinear SDP index Given: polynomials A problem that will arise in this talk is: find such that This is a nonconvex SDP, namely a bilinear matrix inequality

Estimating Region of Attraction Dynamics, equilibrium point If there exists positive-definite V with then (Lyapunov): the set and in the region-of-attraction. is invariant, Strategy: Grow estimate via optimization, enforcing containments with S-procedure/SOS conditions index

Estimating Region of Attraction index Dynamics, equilibrium point p: User-defined function whose sub-level sets are to be in region-of-attraction By choice of positive-definite V, maximize so that: Lyapunov The set is invariant, and in the region -of-attraction.

Convexity of Analysis In a global stability analysis, the certifying Lyapunov functions are themselves a convex set. In local analysis, the condition holds on sublevel sets This set of certifying Lyapunov functions is not convex. Example: index

Checking Set Containments Using SOS index Consider sets defined via inequalities, using given polynomials p. A and p. B as Question: Does the set containment constraint If there exists a polynomial such that then Linearly parametrize class to search (for s) over as Find so hold?

Region of Attraction: Bilinear SOS Maximize (positive-definite V ) so that Simple Psatz: Choose “small” positive definite functions SDP iteration or direct BMI solver (eg. PENBMI from PENOPT) BMIs Products of index

Sanity check index For a positive definite matrix B, nth order system • cubic vector field • known ROA Proof: Consider p. d. quadratic shape factor The best obtainable result is the “largest” value such that That containment easy to characterize: Questions: – Can the bilinear SOS formulation yield this? Yes, s 1=…, s 2=…, s 3=… – Can the BMI solvers find this solution? Basically, Yes, Fast (n<10) But, not all problems work so nicely… 1000’s of random examples, n=2 -10; two restarts of PENBMI, always successful

Other Analysis: Reachability of If w/ inputs Conditions on Conclusion on ODE then Simple Psatz certification BMI index

index gain of If then elementary sufficient condition

Common features of analysis These analysis all involve search over a nonconvex set of certifying Lyapunov functions, roughly The SOS relaxations are nonconvex as well, e. g. , They are “solved” via – (Ad-hoc) iteration on linear SDPs (Initialization? ? ) – PENBMI, commercial BMI solver from PENOPT index

Collect thoughts Nonconvex problems, nonconvex relaxations. Solution approaches: SOS conditions to verify containments – Parametrize V, parametrize multipliers, solve… • Ad-hoc iterative, based on linear SDPs • Bilinear SDP solvers Behavior: – Initial point can have big effect on end result, e. g. , • Unable to reach a feasible point • Convergence to local optimum What are prospects for generating “good” initial V points? – Easily computable – Promising results index

Estimating Region of Attraction index Dynamics, equilibrium point p: User-defined function whose sub-level sets are to be in region-of-attraction By choice of positive-definite V, maximize so that: Lyapunov The set is invariant, and in the region -of-attraction.

ROA: Simulations constrain suitable V Consider a simpler question. Fix β, is Ad-hoc solution: – run N sims, starting from samples in • If any diverge, then “no” • If all converge, then maybe “yes”, and perhaps the Lyapunov analysis can prove it In this case, how can we use the simulation data? Necessary condition: If V exists to verify, it must be – ≤ 1 on all trajectories – ≥ 0 on all trajectories – Decreasing on all trajectories – Other constraints? ? ? … index

Outer bound on certifying Lyapunov functions After simulations – Collection of convergent trajectories starting in – divergent trajectories starting in Linearly parametrize V, namely The necessary conditions on V are convex constraints on V≤ 1 on convergent trajectories V≥ 0 on all trajectories V decreasing on convergent trajectories Quad(V) is a Lyapunov function for Linear(f) V≥ 1 on divergent trajectories index

Hit & Run: Uniformly sample convex set in Rn index 1. Start with an interior point, w 2. Pick a direction v in Rn, N(0, I) 3. Find tmin and tmax such that w+tv just in set 4. Pick μ, uniformly in [tmin tmax] 5. Next. X = x + μv In Lyapunov coefficient space, get samples: – Assess the ROA that they certify, or… – Use as a seed for • SDP iteration and/or PENBMI Finding [tmin tmax] involves – Several simple 1 -d linear inequalities – A linear matrix inequality for – An SOS program, for Smith, 1984 Operations Research Lovasz, 1999 Math Programming Tempo, Calafiore, Dabbene, Springer

Assessing V: Checking containments Each candidate V certifies a region of attraction Generally, this is solved in two steps – SOS optimization (s 1, s 2) to maximize the level-set condition on V (SDP) Bisect on – SOS optimization (s 3) to maximize the condition on p & V (SDP) Bisect on SDP iteration and/or PENBMI are initialized with these as well index

SDP Iteration index 1. Given positive-definite V, define 2. With V and Assess V 3. Given (s 1, s 2, s 3) , V Update V

Summary: Estimating Region of Attraction index Given a “shape” function p QUESTION: Fix β>0, is Dynamics, equilibrium point Find positive-definite V, with Pragmatic solution: – run N sims, starting from samples in nonconvex constraint on V Then is invariant, and in the region of attraction of , denoted – Collection of convergent trajectories starting in – divergent trajectories starting in Necessary cond: If V exists to verify, need V≤ 1 and decreasing on convergent trajectories V≥ 0 on all trajectories Quad(V) is a Lyapunov function for Linear(f) V≥ 1 on the divergent trajectories Necessary conditions are convex constraints on How can we use the simulation data to aid in the nonconvex search for a certifying V? Each candidate V certifies some ROA Simulations yield Linearly parametrize • If any diverge, then unambiguously “no” • If all converge, then maybe “yes”, perhaps Lyapunov analysis to prove/certify it Sample this convex outerbound for candidate Lyapunov fcns Convex constraints in Lyapunov function coefficient-space Assess in 2 steps, using positivity & sum-ofsquares (SOS) optimization to enforce subsets • SOS optimization (s 1, s 2) to maximize the level-set condition on V • SOS optimization (s 3) to maximize condition on p, V Furthermore, coordinatewise solvers are initialized with these, and further optimization (adjusting V too) be performed.

index Example: Van der Pol: ROA Classical 2 -d system Features: – Unstable limit cycle around origin – One equilibrium point: stable, at origin – Here, we use an elliptical shape factor ROA for Van der Pol 3 2 x 2 1 0 -1 -2 -3 -2. 5 βmax ≈1. 04 -2 -1. 5 -1 -0. 5 0 x 1 0. 5 1 1. 5 2 2. 5

Results: Van der Pol’s oscillator Quadratic shape factor: βmax ≈1. 04 Sims performed from Assess achieved β from 50 samples of outer bound Now seed SDP iteration with these samples index

index Van der Pol’s Summary Unseeded PENBMI Degree(V)=4 Degree(V)=6 Run. Time 30 -45(-300) seconds 900 -3000 seconds Best. Answer, β= 0. 928 1. 034 Percentage 90 30 Seeded SDP Iteration Degree(V)=4 Degree(V)=6 Simulations 10 seconds (100) 20 seconds (200) Form LP/Convex. P 1 second 2 seconds Get feasible point 10 seconds 20 seconds Associate multipliers 2 seconds 5 seconds Seed/Run Iteration 7 seconds 16 seconds TOTAL 35 seconds 63 seconds Additional Point (H&R) 0. 1 seconds 0. 2 seconds Associate multipliers 2 seconds 5 seconds Seed/Run Iteration 7 seconds 16 seconds Best. Answer, β= 0. 930 1. 034 Percentage 100

Level Sets The level sets index

5 -state aircraft example index Aircraft: Short period longitudinal model, pitch axis, with 2 -state dynamic inversion controller Simple form for shape factor: Different Lyapunov function structures Quadratic (βcert=8. 6) pointwise-max quadratics (βcert=8. 6) Quadratic+Quartic (βcert=12. 2) Fully quartic (quadratic + cubic + quartic) • βcert=14. 6 Other approaches have deficiencies – Directly use commercial BMI solver (PENBMI) • βcert=15. 2, but… • 38 hours!!! – SDP iteration from “random” starting point • Use P from • Initialize • 30 iterations, βcert=8. 6 4000 simulations 5 minutes Form LP/Convex. P 3 minutes Get a feasable point 5 minutes Assess answer with V 2 minutes SDP Iterate from V 3 minutes/iteration, 6 iters TOTAL discover divergent trajectories – – 33 minutes Divergent initial condition g vin pro Certified set of convergent initial conditions Disk in 5 -d state space, centered at equilibrium point

Technical Progress: Uncertain Systems index Choose V to maximize so that: Uncertain Dynamics Apriori constraint on uncertainty Consider an equilibrium point does not depend on , i. e. , that Same idea: convert to a bilinear SOS problem: basis for V, SOS multipliers, etc… Why? Linear robustness analysis (µ upper bound) Δ test, for M yields a Lyapunov fcn – V proves stability – is a known LFT (ie. , rational function) of. – Explicit formula for involving terms of M & D. So (implicitly) parameter-dependent Lyapunov functions are used often… other places too Possible Strategy – Leverage the known formula for linear case – Employ simulations as before Problems – Dimensionality • 10 state, 5 parameter, cubic (in x) vector field) would be impossible – 1 very large, bilinear SOS problem – Need to pick basis functions instead of just Obvious (but slow to have) realization: – the variables x and δ are different • can be treated similarly, but… – containments involve x, ptwise in δ Several ways to exploit this. One way… – Conservative solutions V(x) for δ in polytope – Cover with a union of polytopes

index Uncertain Systems: Parameter-Independent V For simplicity, take affine parameter uncertainty polytope in Rm Solve earlier conditions, but enforcing at the vertex values of f. Then is invariant, and in the Robust ROA of. Advantages: a robust ROA, and – V is only a function of x, δ appears only implicitly through the vertices – SOS analysis is only in x variables – Simulations are incorporated as before (vary initial condition and δ) Limitations – Conservative with regard to uncertainty Subdivide Δ Solve separately • Conclusions apply to time-varying parameters, hence… • often conclusions are too weak for time-invariant parameters Δ 2 Δ 1

index Much better: B&B in Uncertainty Space Of course, growth is still exponential in parameters… but – kth local problem uses Vk(x) – Solve conservative problem over subdomain – Local problems are decoupled – Trivial parallelization δ 2 Computation yields a binary tree – decomposes parameter space – certificates at each leaf BTree(k). Analysis. Parameter. Domain Analysis. Vertex. Dynamics Analysis. Lyapunov. Certificate Analysis. SOSCertificates Analysis. Certified. Volume BTree(k). Children Nonconvex parameter-space, and/or coupled parameters – cover with union of polytopes, and refine… δ 1

Uncertain Systems: Examples index 2 -state, non-affine uncertainty, sanity check dynamics have strong dependence on as 2 parameters on 1 -d manifold. Cover with polytope. Solve. Refine to union of polytopes. Solve on each polytope. Intersect ROAs → Robust ROA 1. 5 3 1 subdivisions 2 1 0. 9 0. 8 Certified Robust-Region-of-Attractions 0. 7 35 partitions of [0 1], quadratic and quartic V, 0. 6 quadratic and quartic SOS multipliers for each 0. 5 partition, <2 minutes on 8 -machine PC cluster. 1 d 2 repeat Treat 0. 5 x 2 0. 4 0 0. 3 0. 2 Best literature estimate from Chesi, et. , al. 0. 1 -0. 5 -1 -1. 5 -1 -0. 5 0 x 1 0. 5 1 1. 5 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 d 0. 6 0. 7 0. 8 0. 9 1

5 -state aircraft example index Aircraft: Short period longitudinal model, pitch axis, with 2 -state dynamic inversion controller QUESTION: how should plant/controller states be weighted? Same form for shape factor: Earlier: not-uncertain ( ) – Quadratic, βcert=8. 6 7. 26 Uncertain: solve at 4 vertices – Quadratic, βcert=7. 2 – Fully quartic, βcert=9. 7 11. 2 7. 55 11. 1 7. 25 12. 5 8. 46 12. 0 8. 11 11. 1 7. 78 12. 7 9. 11 – Fully quartic, βcert=14. 6, 15. 2 – Divergent IC, with 11. 6 7. 87 12. 2 8. 71 11. 1 8. 35 Uncertain: 9 subdivisions – Quadratic, βcert=7. 2 • (single point issue) – Fully quartic, βcert=11. 1 Divergent IC, with

5 -state aircraft example, time-delay index Aircraft: Short period longitudinal model, pitch axis, with 2 -state dynamic inversion controller Delay via 1 st order Pade, use T = 0. 25(linearized delay margin) Shape factor: Earlier: no-delay – Quadratic, βcert=8. 6 – Quartic, βcert=14. 2 – Divergent IC, with With delay: – Quadratic, βcert=5. 0 – Quartic, βcert=9. 0 – Divergent IC, with Aside: checking SOS of degree=6 polynomial in 6 variables: 84 -by-84 SDP, with 2600+ decision variables

5 -state aircraft example, time-delay, uncertainty index Aircraft: Short period longitudinal model, pitch axis, with 2 -state dynamic inversion controller Set delay as 1 st order Pade, 0. 25 of linearized delay margin Shape Factor Nominal: βcert(∂V=2)=8. 6, βcert(∂V=4)=14. 2 – Divergent IC, with Uncertain: 9. 74 4. 69 8. 38 4. 45 8. 59 4. 25 9. 17 4. 95 8. 39 4. 70 8. 31 4. 47 8. 78 5. 21 9. 00 4. 93 8. 27 4. 67 βcert(∂V=2)=7. 2, βcert(∂V=4)=11. 1 – Divergent IC, with Time-delay: βcert(∂V=2)=5. 0, βcert(∂V=4)=9. 0 – Divergent IC, with earlier Both (time delay, uncertainty) – Quadratic, βcert=4. 25 – Quartic, βcert=8. 3 – Divergent IC, with

Generalization of covering manifold Given: – polynomial p(δ) in many real variables, – domain Find a polytope that covers the manifold – Tradeoff between number of vertices, and – Excess “volume” in polytope One approach: assume H is a polytope – Find “tightest” linear upper and lower bounds over H index

Generalization of covering manifold One step: – Given H, there is a vector w (depends on H), such that for all c So, cost is a linear function of c. The constraints are more troublesome, but can be imposed using the S-procedure, again. Assume H can be written as a system of polynomial inequalities If there exist SOS (s 1, s 2, …, sk) in Σm, with Then index

Reachability of with inputs Given a differential equation and a positive definite function p, how large can get, knowing Special Case: Controllability grammian gives where index

Reachability of with If inputs Conditions on Conclusion on ODE then Simple Psatz certification BMI index

Reachability of with index inputs Example: 16 Decision variables 14 12 10 Upper Bound b 8 Linearized 6 4 2 0 0 2 4 2 R 6 8 10

Reachability of with inputs index Choose T: Tierno, et. al, 1996 Conditions for stationarity repeat adjust scalar so Note: If f is linear, and p is a p. d. quadratic form, then the iteration is the correct power iteration for the maximum.

Same example, with a Lower bound on Reachability 16 14 12 Upper Bound 10 b Lower Bnd 8 Linearized 6 4 2 0 0 2 4 6 2 R 8 10 index

Upper Bound: Refinement Replace with generally, hk<1 will work Then generally, greater than R 2 index

Upper Bound Refinement: Derivation Suppose from x=0, w leads to at some t>0. Then Equivalently: index

Solving the upper bound refinement Replace index (enforced by) with Hold V fixed from first solution, use m partitions, solve m SDPs

index Effect of Refinement 16 14 Refined Upper Bound 12 Upper Bound 10 b Lower Bnd 8 Linearized 6 4 Using worst-case input from linear analysis 2 0 0 2 4 R 2 6 8 10

gain: Adaptive control example Plant: Controller: with unknown (=2) Properties: Global convergence x 1 to 0, x 2 to θ-dependent equilibrium point, and (in this case) Add input disturbance, compute gain from C P How does adaptation gain affect this? “Adaptive nonlinear control without overparametrization, ” Krstic, Kanellakopoulos , Kokotovic, Systems and Control Letters, vol. 19, pp. 177 -185, 1992 index

index gain of If then elementary sufficient condition Iteration (as before) for stationary points, to yield lower bounds

index Adaptive control Compute/Bound C P from equilibrium, for two values of adaptation gain, Γ=1, 4. Adaptive Control, G = 1 and G = 4 0. 55 L 2 to L 2 gain 0. 5 H∞ norm of the linearization For small w, large adaptation gain gives better worst-case disturbance attenuation. Γ=1 0. 45 0. 4 But for large w, the situation is reversed… 0. 35 0. 3 Γ=4 0. 25 0 0. 5 1 R 1. 5 2 Trend implied by linearized analysis invalid for large inputs.

Recap index Questions – Region-of-attraction – Input-to-State, Input-to-Output gain analysis Methods – Lyapunov theory – SOS proofs to ensure set containments – Bilinear SDP to find V – Simulations to seed bilinear optimization Parameter Uncertainty – Cover with affine uncertainty models, polytope values – Single Lyapunov function analysis – Branch-&-Bound to reduce conservatism Accomplishments: heading towards tools that handle – 15 states, 5+ parameters, cubic (in x) vector field, analyze with ∂(V)=2 – 8 states, 3+ parameters, cubic (in x) vector field, analyze with ∂(V)=4 Main Issues – Polynomial models – Dimensionality of SDPs – Applying to validation problems How should this be viewed? Linearized analysis can be thought of as: Infinitesimal analysis of dynamics with quadratic Lyapunov fcns Proposed method extends both the degree of approximation of the dynamics, and the richness of the Lyapunov function • Add insight to the exhaustive linearized analysis • Directly address Mil-Spec goals (Decay rates, Damping ratios, Oscillation freqs, Time-to-double, time delays)

Impact (? ) index Linear analysis provides a quick answer to a related, but different question: Q: How much gain and time-delay variation can we accommodate in flight? A: Here’s a scatter plot of gain margin/time-delay margin at 1000 trim conditions (throughout envelope) Why does linear analysis have impact in nonlinear problems? –Domain-specific expertise exists to interpret linear analysis & assess relevance –Speed, scalable • Fast, defensible answers, on high-dimensional systems Will (these) nonlinear methods ever have such an impact? –Problems where domain knowledge is less well-developed; little/no experience to rely on linear analysis: intuition could break down. • UAVs with unusual airframe designs • Control laws for high-angle-of-attack maneuvers • Adaptive control laws –Problems where performance is being pushed to the limit • approximations associated with linear analysis are no longer good enough.

Transitions Release it – Se. Du. Mi (Sturm, Mc. Master) – SOStools (Antonis, Prajna, Parrilo, Seiler) – Code (UCB/UMN/Seiler) – Polynomial. Dynamical. System class (Wheeler, Seiler) Teach it/Promote it – Website/Wiki for nonlinear analysis problems – Conferences, journals, ½ day workshops Use it Feedback from Industry and National Labs index

Publications 1. 2. 3. 4. 5. 6. 7. 8. index “Stability region analysis using SOS programming, ” 2006 ACC “Local gain analysis of nonlinear systems, ” 2006 ACC “Stability region analysis using simulations and SOS programming, ” 2007 ACC “Stability region analysis via composite Lyapunov functions & SOS programming, ” IEEE TAC, 02/08. “Local stability analysis using simulations & SOS programming, ” under review, Automatica (12/06, 4/07, 12/07) “Stability region analysis for uncertain nonlinear systems, ” to appear 2007 CDC “Local stability analysis for uncertain nonlinear systems, ” submitted IEEE TAC (6/07) “B&B for Local Stability analysis of uncertain nonlinear systems, ” under review, 2008 ACC Project Website http: //jagger. me. berkeley. edu/~pack/certify All examples, certificates, … http: //jagger. me. berkeley. edu/~pack/certificates

1 -unit course: Lecture Schedule 1. Intro: polynomial dynamical systems 2. Lyapunov theorem for ROA 3. ROA: simulations constrain V 4. Review, recap LP, SDP optimization problems 5. SOS→PSD, checking SOS ↔ SDP 6. Containments, empty intersections, as SOS, general Psatz 7. Applying 1 -6 to ROA problems 8. Code, tools, results for 7 9. Disturbance-to-state problems 10. Uncertain Dynamics, vertex results, Branch and Bound 11. Covering nonpolynomial systems with poly model, error 12. Global & local Control Lyapunov Function (CLF) synthesis 13. Other literature, approaches (Lall, Glavski, Prajna, etc) 14. Other literature (continued) index